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Full text of "Treatise On Analysis Vol-Ii"

1    DEFINITION OF A MEASURE        101

a linear form on tf (X). It is a measure, for if K is any compact subset
of X and if /e Jf (X; K), then \gf\  \\f\\  sup \g(x)\9 and consequently

I ^ bK II/H, where fcK = % sup |#(*)|. This measure is denoted by g 

xeK

and it is called the measure with density g relative to ju (cf. (13.13)).

(13.1.6) Let TC : X-X' be a homeomorphism of X onto a locally compact
space X'. For each function /e ?f(X'), the function / n belongs to ^f(X),
and we have Supp(/o n) = n"1(Suppf). It follows immediately that, if \L is
any measure on X, then/W/i(/o 71) is a measure on X', called the zma#e of /i
under TC and denoted by 7t(/j).

(1 3,1 .7) Let Y be a closed subset of X (and therefore a locally compact sub-
space of X (3.18.4)) and v a measure on Y. For each / e Jf (X; K), the
restriction /| Y belongs to Jf(Y; K n Y), and hence there exists a constant
CK such that

yeY

for all/e ^(X; K). The mapping /V-v(/|Y) is therefore a measure on X,
called the *>wa# of v under the canonical injection Y -> X, or the canonical
extension ofv to X.

(1 3.1 .8) Let U be an open subset of X (and therefore again a locally compact
subspace (3.18.4)). For each compact subset K of U, it is clear that the
mapping/W/| U is an isometry of Jf (X; K) onto tf (U ; K). The image under
the inverse isometry of a function ge Jf(U; K) is the function #u which
agrees with g on U and is zero on X - U. (By abuse of notation, we shall
often write /in place of/| U when Supp(/) c U, and g in place of #u). The
mapping g^g" of Jfc(U) into ^fc(X) is therefore injective. If /* is any
measure on X, the mapping g h- X0U) is a measure on U, said to be induced by
ju OT U, or the restriction of p. to U, and denoted by ^ or ju | U. It should be
noted that a measure v on U is not necessarily the restriction of a measure on
X (Section (13.4), Problem 1), and that an "extension" of v to X, if it exists,
is not in general unique. However, there is the following result:

(13.1.9) Let (Ua)a6l be an open covering ofX. For each a el let ua be a
measure on Ua such that, for each pair of indices a, j8, the restrictions of i*a
and Up to Ua n U^ (13.1 .8) are equal. Then there exists a unique measure \i on
X whose restriction to Ua is na,for each a 6 1.

We shall first show that each function /e tf C(X) can be written in the
form/= /, where, for each index i, there exists a( e I such that/f e Jfc(X)ntable set of points of discontinuity of / is dense in [a, b[ (Section 3.15,